The small $p$-adic Simpson comparison conjecture for semi-stable formal schemes

Let ${\mathfrak X}_S$ be a liftable semi-stable formal scheme over $A^+$ with a fixed lifting $\widetilde {\mathfrak X}_S$ over ${{\mathbb A}_{2,K}}(S)$. Let ${\mathcal M}$ be a Hitchin-small integral $v$-bundle on $X_{S,v}$, and let $({\mathcal H}^+,\theta)$ be its associated twisted Hitchin-small Higgs bundle. Denote by $r$ the nilpotency length of $(\zeta_p-1)\theta$ modulo $p$. The canonical morphism

$${\mathrm D}{\mathrm R}({\mathcal H}^+,\theta)\to {\mathrm R}\nu_*{\mathcal M}^+$$

Small $p$-adic Simpson comparison conjecture. The canonical morphism induces a quasi-isomorphism

$$\tau^{\leq p-r}{\mathrm D}{\mathrm R}({\mathcal H}^+,\theta)\simeq \tau^{\leq p-r}{\mathrm L}\eta_{\rho_K(\zeta_p-1)}{\mathrm R}\nu_*{\mathcal M}^+.$$

This conjecture predicts a truncated comparison between the de Rham complex of the associated Higgs bundle and the derived pushforward of the corresponding $v$-bundle after applying the décalage functor. It extends the preceding decomposition result from the structure sheaf to Hitchin-small integral $v$-bundles; its resolution would provide a broad semi-stable integral form of the small $p$-adic Simpson correspondence.